Optimal. Leaf size=140 \[ -\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+x}-x\right )}{4 \sqrt [3]{2}}-\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+x}+x}\right )}{4 \sqrt [3]{2}}-\frac {3 \sqrt [3]{x^3+x} \left (2 x^2-1\right )}{4 x^3}+\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+x} x+2^{2/3} \left (x^3+x\right )^{2/3}+x^2\right )}{8 \sqrt [3]{2}} \]
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Rubi [A] time = 0.38, antiderivative size = 228, normalized size of antiderivative = 1.63, number of steps used = 13, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2056, 580, 583, 12, 466, 465, 494, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {3 \sqrt [3]{x^3+x}}{2 x}+\frac {3 \sqrt [3]{x^3+x}}{4 x^3}-\frac {3 \sqrt [3]{x^3+x} \log \left (\sqrt [3]{2}-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{x^2+1}}+\frac {3 \sqrt [3]{x^3+x} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {\sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2+1}}+2^{2/3}\right )}{8 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{x^2+1}}-\frac {3 \sqrt {3} \sqrt [3]{x^3+x} \tan ^{-1}\left (\frac {\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 204
Rule 292
Rule 465
Rule 466
Rule 494
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (-4+x^2\right ) \sqrt [3]{x+x^3}}{x^4 \left (2+x^2\right )} \, dx &=\frac {\sqrt [3]{x+x^3} \int \frac {\left (-4+x^2\right ) \sqrt [3]{1+x^2}}{x^{11/3} \left (2+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}+\frac {\left (3 \sqrt [3]{x+x^3}\right ) \int \frac {\frac {32}{3}+\frac {40 x^2}{3}}{x^{5/3} \left (1+x^2\right )^{2/3} \left (2+x^2\right )} \, dx}{16 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {\left (9 \sqrt [3]{x+x^3}\right ) \int -\frac {32 \sqrt [3]{x}}{3 \left (1+x^2\right )^{2/3} \left (2+x^2\right )} \, dx}{64 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3} \left (2+x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (9 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3} \left (2+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (9 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3} \left (2+x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (9 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{2-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left (3 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {3 \sqrt [3]{x+x^3} \log \left (\sqrt [3]{2}-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left (9 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {3 \sqrt [3]{x+x^3} \log \left (\sqrt [3]{2}-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {3 \sqrt [3]{x+x^3} \log \left (2^{2/3}+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {\sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (9 \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3 \sqrt [3]{x+x^3}}{4 x^3}-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {3 \sqrt {3} \sqrt [3]{x+x^3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {3 \sqrt [3]{x+x^3} \log \left (\sqrt [3]{2}-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {3 \sqrt [3]{x+x^3} \log \left (2^{2/3}+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {\sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt [3]{2} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 102, normalized size = 0.73 \begin {gather*} \frac {3 \sqrt [3]{x^3+x} \left (9 \left (x^2+2\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {x^2}{2 x^2+2}\right )+\left (9 x^4-6 x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x^2}{2 x^2+2}\right )-4 \left (11 x^4+7 x^2-4\right )\right )}{64 x^3 \left (x^2+1\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.41, size = 140, normalized size = 1.00 \begin {gather*} -\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+x}-x\right )}{4 \sqrt [3]{2}}-\frac {3 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+x}+x}\right )}{4 \sqrt [3]{2}}-\frac {3 \sqrt [3]{x^3+x} \left (2 x^2-1\right )}{4 x^3}+\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+x} x+2^{2/3} \left (x^3+x\right )^{2/3}+x^2\right )}{8 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.11, size = 288, normalized size = 2.06 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{4} + 5 \, x^{2} + 2\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}} - 12 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{5} + 22 \, x^{3} + 4 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} - 2^{\frac {5}{6}} {\left (91 \, x^{6} + 168 \, x^{4} + 84 \, x^{2} + 8\right )}\right )}}{6 \, {\left (53 \, x^{6} + 48 \, x^{4} - 12 \, x^{2} - 8\right )}}\right ) - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + x\right )}^{\frac {2}{3}} {\left (2 \, x^{2} + 1\right )} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{4} + 22 \, x^{2} + 4\right )} + 6 \, {\left (5 \, x^{3} + 4 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{2} + 4}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 2\right )} + 6 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{x^{2} + 2}\right ) - 12 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{2} - 1\right )}}{16 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 97, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + \frac {3}{16} \cdot 4^{\frac {1}{3}} \log \left (\left (\frac {1}{2}\right )^{\frac {2}{3}} + \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right ) - \frac {3}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {1}{2}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {9}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.74, size = 1587, normalized size = 11.34 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3 \, {\left (18 \, x^{5} + 7 \, {\left (x^{3} + x\right )} x^{2} + 8 \, x^{3} - 10 \, x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{56 \, {\left (x^{\frac {17}{3}} + 2 \, x^{\frac {11}{3}}\right )}} + \int \frac {9 \, {\left (3 \, x^{4} - x^{2} - 4\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{7 \, {\left (x^{\frac {23}{3}} + 4 \, x^{\frac {17}{3}} + 4 \, x^{\frac {11}{3}}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-4\right )\,{\left (x^3+x\right )}^{1/3}}{x^4\,\left (x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 2\right ) \left (x + 2\right )}{x^{4} \left (x^{2} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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